Introduction
The equation of a quadratic function can be written in
several different forms. We have practiced using the
standard form of a quadratic function to identify key
information and graph the function. In this lesson, we will
explore writing quadratic functions in intercept form or
factored form, graphing a quadratic function from
intercept form, and creating the equation of a quadratic
function given its graph. An equation written in intercept
form allows you to easily identify the x-intercepts of a
quadratic function.
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Key Concepts
• A quadratic function in standard form can be created
from the intercept form.
• The intercept form of a quadratic function is
f(x) = a(x – p)(x – q), where p and q are the zeros of
the function.
• To write a quadratic in intercept form, distribute the
terms of the first set of parentheses over the terms of
the second set of parentheses and simplify.
• For example, f(x) = 3(x – 4)(x – 1) becomes
f(x) = 3x2 – 15x + 12.
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Key Concepts, continued
• The y-intercept of a quadratic function written in
intercept form can be found by substituting 0 for x.
• The x-intercept(s) of a quadratic function written in
intercept form can be found by substituting 0 for y.
• The x-intercepts, also known as the zeros, roots, or
solutions of the quadratic, are often identified as
p and q.
• The axis of symmetry is located halfway between the
zeros. To determine its equation, use the formula
p+q
x=
.
2
5.3.2: Creating and Graphing Equations Using the x-intercepts
3
Key Concepts, continued
• Recall that the axis of symmetry extends through the
vertex of a quadratic function.
• As such, the x-coordinate of the vertex is the value of
the equation of the axis of symmetry.
• If the vertex is the point (h, k), then the equation of the
axis of symmetry is x = h.
• The y-value can be found by substituting the x-value
into the equation.
4
5.3.2: Creating and Graphing Equations Using the x-intercepts
Common Errors/Misconceptions
• multiplying out the factored form when it is more
advantageous not to
• drawing a parabola as a straight line or a series of lines
instead of a curve
5
5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice
Example 2
Identify the x-intercepts, the axis of symmetry, and the
vertex of the quadratic f(x) = (x + 5)(x + 2). Use this
information to graph the function.
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 2, continued
1. Identify the x-intercepts and plot the
points.
The x-intercepts of the quadratic are the zeros or
solutions of the quadratic.
Start by setting the equation equal to 0 and solving
to determine the zeros.
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 2, continued
f(x) = (x + 5)(x + 2)
Original equation
0 = (x + 5)(x + 2)
Set the equation
equal to 0.
0 = x + 5 or 0 = x + 2
Set each factor equal
to 0.
–5 = x or –2 = x
Solve each equation
for x.
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 2, continued
2. Determine the axis of symmetry to find the
vertex.
The axis of symmetry is the line that divides the
parabola in half.
To determine its equation, find the midpoint of the two
zeros and draw the vertical line through that point.
p+q
Insert the values of x into the formula x =
to find
2
the midpoint.
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 2, continued
Let p = –5 and q = –2.
x=
-5 + ( -2)
2
=
-7
2
= -3.5
The equation of the axis of symmetry is x = –3.5.
The axis of symmetry extends through the vertex of
the quadratic. This means that the vertex (h, k) of the
graph has an x-value of –3.5.
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 2, continued
Substitute –3.5 for x in the original equation to
determine the y-value of the vertex.
f(x) = (x + 5)(x + 2)
Original equation
f(–3.5) = (–3.5 + 5)(–3.5 + 2)
Substitute –3.5
for x.
f(–3.5) = (1.5)(–1.5)
Simplify.
f(–3.5) = –2.25
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 2, continued
The y-value of the vertex is –2.25.
The vertex is the point (–3.5, –2.25).
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 2, continued
3. Graph the zeros, the vertex, and the axis
of symmetry of the given equation.
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 2, continued
4. Sketch the function based on the zeros
and the vertex.
✔
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 2, continued
15
5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice
Example 3
Use the intercepts and
a point on the graph at
right to write the
equation of the function
in standard form.
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 3, continued
1. Find the x-intercepts of the graph and
write them as zeros of the equation.
Identify the two points of intersection and write them
as zeros to prepare to write the factors of the
equation.
The x-intercepts are (–2, 0) and (3, 0).
The zeros are x = –2 and x = 3.
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 3, continued
2. Write the zeros as expressions equal to 0
to determine the factors of the quadratic.
x = –2
x+2=0
x=3
x–3=0
The factors of the equation are (x + 2) and (x – 3), so
the equation is f(x) = a(x + 2)(x – 3).
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 3, continued
3. Use the point (4, –3) to find a, the
coefficient of the x2 term.
The equation is f(x) = a(x + 2)(x – 3) is written in
intercept form, or f(x) = a(x – p)(x – q).
Substitute (4, –3) into the equation to find a.
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 3, continued
f(x) = a(x + 2)(x – 3)
Equation
–3 = a(4 + 2)(4 – 3)
Substitute (4, –3) for x
and f(x).
–3 = a(6)(1)
Simplify.
–3 = 6a
1
- =a
2
The coefficient of
x2
1
is - .
2
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 3, continued
4. Write the equation in standard form.
Insert the value found for a into the equation from
step 1 and solve.
f(x) = a(x + 2)(x – 3)
1
f (x) = - (x + 2)(x - 3)
2
Equation
Substitute -
1
2
for a.
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 3, continued
1
f (x) = - (x 2 - x - 6)
2
1
1
f (x) = - x 2 + x + 3
2
2
The coefficient of
x2
Simplify by multiplying
the factors.
Distribute -
1
2
parentheses.
1
1
over the
is f (x) = - x + x + 3.
2
2
2
✔
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5.3.2: Creating and Graphing Equations Using the x-intercepts
Guided Practice: Example 3, continued
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5.3.2: Creating and Graphing Equations Using the x-intercepts